A Hybrid Phase Flow Method for solving Liouville Equation in Bounded Domain∗

The phase flow method was originally introduced in [28] which can efficiently solve autonomous ordinary differential equations. In [13], the method was generalized to solve Hamiltonian system where the Hamiltonian function was discontinuous. However, both these methods require phase flow map constructed on an invariant manifold. This can increase computational cost when the invariant domain is big or unbounded. Following the idea of [13], we propose a hybrid phase flow method for solving Liouville equation in bounded domain, which is smaller than the invariant manifold of phase flow map. By using some proper boundary conditions, this method can help solve the problem where the invariant manifold of phase flow map determined by Liouville equation is unbounded. We verify numerical accuracy and efficiency by several examples of the semiclassical limit of Schrödinger equation. Analysis of numerical stability and convergence is given for the semiclassical limit equation with inflow boundary condition.